HW_2_Key - Miller, Kierste Homework 2 Due: Sep 6 2007, 3:00...

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Miller, Kierste – Homework 2 – Due: Sep 6 2007, 3:00 am – Inst: JEGilbert 1 This print-out should have 17 questions. Multiple-choice questions may continue on the next column or page – fnd all choices beFore answering. The due time is Central time. YES, homework 2 is due BEFORE homework 1 001 (part 1 oF 1) 10 points Determine A so that the curve y = 7 x + 17 can be written in parametric Form as x ( t ) = t - 3 , y ( t ) = At - 4 . 1. A = 9 2. A = - 7 3. A = 8 4. A = - 9 5. A = - 8 6. A = 7 correct Explanation: We have to eliminate t From the parametric equations For x and y . Now From the equation For x it Follows that t = x + 3. Thus y = 7 x + 17 = A ( x + 3) - 4 . Consequently A = 7 . keywords: Cartesian equation, parametric equations, eliminate parameter 002 (part 1 oF 1) 10 points Determine a Cartesian equation For the curve given in parametric Form by x ( t ) = 4 e 2 t , y ( t ) = 3 e - t . 1. x y 2 = 48 2. xy 2 = 48 3. xy 2 = 36 correct 4. x y 2 = 36 5. x 2 y = 12 6. x 2 y = 12 Explanation: We have to eliminate the parameter t From the equations For x and y . Now From the equation For x it Follows that e t = x 4 · 1 / 2 , ±rom which in turn it Follows that y = 3 4 x · 1 / 2 . Consequently, x 2 y = 36 . keywords: Cartesian equation, parametric equations, eliminate parameter 003 (part 1 oF 1) 10 points Describe the motion oF a particle with posi- tion P ( x, y ) when x = 5 sin t, y = 3 cos t as t varies in the interval 0 t 2 π . 1. Moves once clockwise along the ellipse x 2 25 + y 2 9 = 1 , starting and ending at (0 , 3). correct
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Miller, Kierste – Homework 2 – Due: Sep 6 2007, 3:00 am – Inst: JEGilbert 2 2. Moves once clockwise along the circle (5 x ) 2 + (3 y ) 2 = 1 , starting and ending at (0 , 3). 3. Moves once counterclockwise along the ellipse x 2 25 + y 2 9 = 1 , starting and ending at (0 , 3). 4. Moves along the line x 5 + y 3 = 1 , starting at (5 , 0) and ending at (0 , 3). 5. Moves along the line x 5 + y 3 = 1 , starting at (0 , 3) and ending at (5 , 0). 6. Moves once counterclockwise along the circle (5 x ) 2 + (3 y ) 2 = 1 , starting and ending at (0 , 3). Explanation: Since cos 2 t + sin 2 t = 1 for all t , the particle travels along the curve given in Cartesian form by x 2 25 + y 2 9 = 1 ; this is an ellipse centered at the origin. At t = 0, the particle is at (5 sin 0 , 3 cos 0), i.e. , at the point (0 , 3) on the ellipse. Now as t increases from t = 0 to t = π/ 2, x ( t ) increases from x = 0 to x = 5, while y ( t ) decreases from y = 3 to y = 0 ; in particular, the particle moves from a point on the positive y -axis to a point on the positive x -axis, so it is moving clockwise . In the same way, we see that as t increases from π/ 2 to π , the particle moves to a point on the negative y -axis, then to a point on the negative x -axis as t increases from π to 3 π/ 2, until Fnally it returns to its starting point on the positive y -axis as t increases from 3 π/ 2 to 2 π .
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This note was uploaded on 05/08/2008 for the course M 408 M taught by Professor Gilbert during the Fall '07 term at University of Texas at Austin.

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HW_2_Key - Miller, Kierste Homework 2 Due: Sep 6 2007, 3:00...

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